# American Institute of Mathematical Sciences

• Previous Article
Non-autonomous weakly damped plate model on time-dependent domains
• DCDS-S Home
• This Issue
• Next Article
Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $\mathbb R^N$$^\diamondsuit$
September  2021, 14(9): 3305-3318. doi: 10.3934/dcdss.2021080

## $W^{2, p}$-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

 1 School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei 050016, China 2 Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China 3 School of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050016, China

* Corresponding author: Shenzhou Zheng

Received  March 2020 Revised  January 2021 Published  September 2021 Early access  June 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (No. 12001160), Natural Science Foundation of Hebei Province (No. A2019205218), Science Foundation of Hebei Normal University (No. L2019B02). The second author is supported by National Natural Science Foundation of China (No. 12071021)

We prove a global $W^{2, p}$-estimate for the viscosity solution to fully nonlinear elliptic equations $F(x, u, Du, D^{2}u) = f(x)$ with oblique boundary condition in a bounded $C^{2, \alpha}$-domain for every $\alpha\in (0, 1)$. Here, the nonlinearities $F$ is assumed to be asymptotically $\delta$-regular to an operator $G$ that is $(\delta, R)$-vanishing with respect to $x$. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global $W^{2, p}$-estimate for the viscosity solution to fully nonlinear parabolic equations $F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t)$ with oblique boundary condition in a bounded $C^{3}$-domain.

Citation: Junjie Zhang, Shenzhou Zheng, Chunyan Zuo. $W^{2, p}$-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3305-3318. doi: 10.3934/dcdss.2021080
##### References:
 [1] T. Alberico, C. Capozzoli, R. Schiattarella and L. D'Onofrio, G-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 129-137.  doi: 10.3934/dcdss.2019009.  Google Scholar [2] S.-S. Byun and J. Han, $W^{2, p}$-estimates for fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations, 268 (2020), 2125-2150.  doi: 10.1016/j.jde.2019.09.018.  Google Scholar [3] S.-S. Byun and J. Han, $L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, arXiv: 2012.07435v1 [math.AP], 43 pages. Google Scholar [4] S.-S. Byun, M. Lee and D. K. Palagachev, Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260 (2016), 4550-4571.  doi: 10.1016/j.jde.2015.11.025.  Google Scholar [5] S.-S. Byun, J. Oh and L. Wang, Global Calderón-Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN, 2015 (2015), 8289-8308.  doi: 10.1093/imrn/rnu203.  Google Scholar [6] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213.  doi: 10.2307/1971480.  Google Scholar [7] L. A. Caffarelli and Q. Huang, Estimates in the generalized Campamato-John-Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., 118 (2003), 1-17.  doi: 10.1215/S0012-7094-03-11811-6.  Google Scholar [8] M. Chipot and L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 291-303.  doi: 10.1017/S0308210500026378.  Google Scholar [9] J. Choi, H. Dong and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374.  doi: 10.3934/dcds.2018097.  Google Scholar [10] H. Dong and N. V. Krylov, On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions, Commun. Partial Diff. Equ., 38 (2013), 1038-1068.  doi: 10.1080/03605302.2012.756013.  Google Scholar [11] H. Dong, N. V. Krylov and X. Li, On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains, St. Petersburg Math. J., 24 (2013), 39-69.  doi: 10.1090/S1061-0022-2012-01231-8.  Google Scholar [12] L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423.  doi: 10.1512/iumj.1993.42.42019.  Google Scholar [13] M. Foss, Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura Appl., 187 (2008), 263-321.  doi: 10.1007/s10231-007-0045-2.  Google Scholar [14] C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully nonlinear elliptic equations, Advances in Mathematics, 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.  Google Scholar [15] N. V. Krylov, Linear and fully nonlinear elliptic equations with $L_{d}$-drift, Commun. Partial Diff. Equ., 45 (2020), 1778-1798.  doi: 10.1080/03605302.2020.1805462.  Google Scholar [16] T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427. doi: 10.1016/j.matpur.2012.02.004.  Google Scholar [17] D. Li and K. Zhang, Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228 (2018), 923-967.  doi: 10.1007/s00205-017-1209-x.  Google Scholar [18] S. Liang and S. Zheng, Calderón-Zygmund estimate for asymptotically regular non-uniformly elliptic equations, J. Math. Anal. Appl., 484 (2020), 123749, 17 pp. doi: 10.1016/j.jmaa.2019.123749.  Google Scholar [19] P. Marcellini, Regularity under general and $p, q$-growth conditions, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2009-2031.  doi: 10.3934/dcdss.2020155.  Google Scholar [20] E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Commun. Partial Diff. Equ., 31 (2006), 1227-1252.  doi: 10.1080/03605300600634999.  Google Scholar [21] C. Scheven and T. Schmidt, Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 469-507.  doi: 10.2422/2036-2145.2009.3.04.  Google Scholar [22] L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.  doi: 10.1002/cpa.3160450103.  Google Scholar [23] N. Winter, $W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164.  doi: 10.4171/ZAA/1377.  Google Scholar [24] J. Zhang and S. Zheng, Lorentz estimates for asymptotically regular fully nonlinear parabolic equations, Math. Nachr., 291 (2018), 996-1008.  doi: 10.1002/mana.201600497.  Google Scholar [25] J. Zhang, M. Cai and S. Zheng, Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type, Nonlinear Anal., 180 (2019), 225-235.  doi: 10.1016/j.na.2018.10.013.  Google Scholar

show all references

##### References:
 [1] T. Alberico, C. Capozzoli, R. Schiattarella and L. D'Onofrio, G-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 129-137.  doi: 10.3934/dcdss.2019009.  Google Scholar [2] S.-S. Byun and J. Han, $W^{2, p}$-estimates for fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations, 268 (2020), 2125-2150.  doi: 10.1016/j.jde.2019.09.018.  Google Scholar [3] S.-S. Byun and J. Han, $L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, arXiv: 2012.07435v1 [math.AP], 43 pages. Google Scholar [4] S.-S. Byun, M. Lee and D. K. Palagachev, Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260 (2016), 4550-4571.  doi: 10.1016/j.jde.2015.11.025.  Google Scholar [5] S.-S. Byun, J. Oh and L. Wang, Global Calderón-Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN, 2015 (2015), 8289-8308.  doi: 10.1093/imrn/rnu203.  Google Scholar [6] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213.  doi: 10.2307/1971480.  Google Scholar [7] L. A. Caffarelli and Q. Huang, Estimates in the generalized Campamato-John-Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., 118 (2003), 1-17.  doi: 10.1215/S0012-7094-03-11811-6.  Google Scholar [8] M. Chipot and L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 291-303.  doi: 10.1017/S0308210500026378.  Google Scholar [9] J. Choi, H. Dong and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374.  doi: 10.3934/dcds.2018097.  Google Scholar [10] H. Dong and N. V. Krylov, On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions, Commun. Partial Diff. Equ., 38 (2013), 1038-1068.  doi: 10.1080/03605302.2012.756013.  Google Scholar [11] H. Dong, N. V. Krylov and X. Li, On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains, St. Petersburg Math. J., 24 (2013), 39-69.  doi: 10.1090/S1061-0022-2012-01231-8.  Google Scholar [12] L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423.  doi: 10.1512/iumj.1993.42.42019.  Google Scholar [13] M. Foss, Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura Appl., 187 (2008), 263-321.  doi: 10.1007/s10231-007-0045-2.  Google Scholar [14] C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully nonlinear elliptic equations, Advances in Mathematics, 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.  Google Scholar [15] N. V. Krylov, Linear and fully nonlinear elliptic equations with $L_{d}$-drift, Commun. Partial Diff. Equ., 45 (2020), 1778-1798.  doi: 10.1080/03605302.2020.1805462.  Google Scholar [16] T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427. doi: 10.1016/j.matpur.2012.02.004.  Google Scholar [17] D. Li and K. Zhang, Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228 (2018), 923-967.  doi: 10.1007/s00205-017-1209-x.  Google Scholar [18] S. Liang and S. Zheng, Calderón-Zygmund estimate for asymptotically regular non-uniformly elliptic equations, J. Math. Anal. Appl., 484 (2020), 123749, 17 pp. doi: 10.1016/j.jmaa.2019.123749.  Google Scholar [19] P. Marcellini, Regularity under general and $p, q$-growth conditions, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2009-2031.  doi: 10.3934/dcdss.2020155.  Google Scholar [20] E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Commun. Partial Diff. Equ., 31 (2006), 1227-1252.  doi: 10.1080/03605300600634999.  Google Scholar [21] C. Scheven and T. Schmidt, Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 469-507.  doi: 10.2422/2036-2145.2009.3.04.  Google Scholar [22] L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.  doi: 10.1002/cpa.3160450103.  Google Scholar [23] N. Winter, $W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164.  doi: 10.4171/ZAA/1377.  Google Scholar [24] J. Zhang and S. Zheng, Lorentz estimates for asymptotically regular fully nonlinear parabolic equations, Math. Nachr., 291 (2018), 996-1008.  doi: 10.1002/mana.201600497.  Google Scholar [25] J. Zhang, M. Cai and S. Zheng, Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type, Nonlinear Anal., 180 (2019), 225-235.  doi: 10.1016/j.na.2018.10.013.  Google Scholar
 [1] Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675 [2] K. T. Joseph, Manas R. Sahoo. Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2091-2118. doi: 10.3934/cpaa.2013.12.2091 [3] Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627 [4] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [5] R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 [6] Mahamadi Warma. Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1881-1905. doi: 10.3934/cpaa.2013.12.1881 [7] Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601 [8] Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 [9] G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279 [10] Patrick Winkert. Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (2) : 785-802. doi: 10.3934/cpaa.2013.12.785 [11] Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070 [12] Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861 [13] Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285 [14] Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure & Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043 [15] Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763 [16] Sami Aouaoui, Rahma Jlel. On some elliptic equation in the whole euclidean space $\mathbb{R}^2$ with nonlinearities having new exponential growth condition. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4771-4796. doi: 10.3934/cpaa.2020211 [17] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [18] Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 [19] Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253 [20] Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

2020 Impact Factor: 2.425